FUNCTIONAL INEQUALITIES FOR SPECTRAL RADII OF NONNEGATIVE MATRICES

Let P+n denote the set of all square n × n nonnegative matrices. For Ak= (akij)ni,j=1, k=1,..., m (k not a power), we set f(A1,..., Am) = (f(a1ij,..., amij))ni,j = 1. For each A ε{lunate} P+n, we let ρ(A) denote its spectral radius. This paper is concerned with the characterization of those functions f:Rm+→R+ satisfying either ρ(f(A1,..., Am))≤f(ρ(A1),..., ρ(Am)) (1) or f(ρ(A1),..., ρ(Am))≤ρ(f(A1,..., Am)) (2) for all A1,..., Amε{lunate}P+n and every n ε N. We totally characterize all functions satisfying (1). We delineate various classes of functions which satisfy (2). If f(0)=0, and f is bounded above in some neighborhood of any αεintRm+, then we totally characterize all f satisfying (2). © 1990.